Network location and clustering of genetic mutations determine chronicity in a stylized model of genetic diseases

In a highly simplified view, a disease can be seen as the phenotype emerging from the interplay of genetic predisposition and fluctuating environmental stimuli. We formalize this situation in a minimal model, where a network (representing cellular regulation) serves as an interface between an input layer (representing environment) and an output layer (representing functional phenotype). Genetic predisposition for a disease is represented as a loss of function of some network nodes. Reduced, but non-zero, output indicates disease. The simplicity of this genetic disease model and its deep relationship to percolation theory allows us to understand the interplay between disease, network topology and the location and clusters of affected network nodes. We find that our model generates two different characteristics of diseases, which can be interpreted as chronic and acute diseases. In its stylized form, our model provides a new view on the relationship between genetic mutations and the type and severity of a disease.

The second method evaluates each (reaction) node together with its next-to-nearest neighbours. In this sense, the method is closer to the nature of nodes in our model, which do not represent individual reactions, but rather more complex regulatory entities, summarizing metabolic flow and genetic control. Such subgraph objects may behave like AND or OR gate depending on the mutual proportions of the alternative sources and the number of reactants.
This method utilizes information about reaction network including their directionality. Each reaction has a form of R 1 + R 2 + ... → P 1 + P 2 + ..., where R i are reactants and P j products. Our quantification focuses on reactants alone. Each reactant may have one or more possible sources, where the number of these sources for each reactant is k i . Hence, the reaction itself corresponds to a Boolean AND, but alternative sourcing is an analogue of a Boolean OR. Such a subgraph has the structure of multiple logical ORs fed to an single logical AND: (R 1,1 |R 1,2 |...) + (R 2,1 |R 2,2 |...) + ... → P 1 + P 2 + .... In order to estimate the parameter a, we need to convert this whole entity to just a single AND or OR.
All fluxes are regarded to be discrete, hence inputs can be characterized by a probability c in of having reactant i from the source j, R i, j . With this quantity, the probability of reaction taking place, c out , can be computed. For a logical AND c out < c in whereas for a logical OR c out > c in , there are also special cases of c in = 0 and c in = 1. The same classification can be done for more complicated structures, like the one discussed above. For simple gates, there is always (for each c in ) either c out < c in or c out > c in depending on the gate. However, for more complicated entities this inequality may change sign upon change of c in .
Way to overcome this difficulty is to compare whole range of c in ∈ [0, 1] by taking integral and then comparing: where i is the metabolite index and k i is the number of alternative sources for this metabolite. Reactions with empty sets of sources were removed from the analysis. Then if α > 0 node is regarded as OR, in the case of α < 0 node is AND. The value of a obtained with this method is between 0.08 and 0.24 for human metabolic models (see Table 1). These values are clearly located in the subcritical regime. This is in line with our model results, which show that only the subcritical regime is viable and resilient to defect and perturbations.
We also checked metabolic models for other living organisms and we found substantial agreement between the values of a. This is a suitable starting point for further investigation. Also, the precise determination of this parameter deserves a more detailed study.